Separable Differential Equations

• Separate the variables.

• Integrate.

	integrate using the power rule
	subtract from both sides

Let Then the equation becomes

• If you are asked to find a particular solution, the next step would be to use
the additional information to solve for C . Since we are finding the general
solution in this problem, our answer will include the variable C.

• Solve for y.

	multiply both sides of the equation by 3
	simplify
	take the cube root of both sides

So the general solution to this differential equation is

where C is any real number .

Use the power rule:

C for n ≠ -1.

• Separate the variables.

• Integrate.

	integrate using the power rule
	subtract from both sides

Let Then the equation becomes

• If you are asked to find a particular solution, the next step would be to use
the additional information to solve for C. Since we are finding the general
solution in this problem, our answer will include the variable C.

• Solve for y.

	multiply both sides of the equation by -3
	simplify
	raise both sides to -1/3
	use the law of exponents

So the general solution to this differential equation is

where C is any real number.

• Separate the variables.

We want to use the formulas and
so we need to change the square root signs into fractional exponents , and use
the laws of exponents to move the e^4x into the numerator .

• Integrate.

	subtract from both sides

Let Then the equation becomes

• If you are asked to find a particular solution, the next step would be to use
the additional information to solve for C. Since we are finding the general
solution in this problem, our answer will include the variable C.

• Solve for y

	multiply both sides of the equation by -4
	simplify
	take the natural log (ln) of both sides
	divide both sides of the equation by -4

So the general solution to this differential equation is

where C is any real number.

• Separate the variables.
First, we will use the law of exponents that says

Now separate the variables.

We want to use the formula so we use the laws of exponents
to move the e^y into the numerator.

• Integrate.

	subtract from both sides

Let Then the equation becomes

• If you are asked to find a particular solution, the next step would be to use
the additional information to solve for C. Since we are finding the general
solution in this problem, our answer will include the variable C.

• Solve for y.

	multiply both sides of the equation by -1
	take the natural log (ln) of both sides
	multiply both sides of the equation by -1

So the general solution to this differential equation is

where C is any real number.

a. and y(0) = 3.
In part (1a) above, we found

Substitute x = 0 and y = 4, and solve for C.

In part (1a) above, we found

so the particular solution to this differential equation with y(0) = 4 is

and y(1) = 8.

In part (1b) above, we found

Substitute x = 1 and y = 8, and solve for C.

In part (1b) above, we found

so the particular solution to this differential equation with y(1) = 8 is